When: ordinary infix maths (real/complex), trig, powers, eng suffixes, angle suffixes, polar ∠. Not for matrices, stats lists, BASE-N, TVM, or unit conversion tables. Params: expression (required); angle_mode=rad|deg|grad (default rad); complex_form=rectangular|polar; variables={name:float}; eng_symbols=bool. Example: expression="sin(30°)", angle_mode="rad"; or "500k+10M"; or "2∠90" with angle_mode="deg".

When: unknown function/operator name after evaluate unknown_token, or exploring arity. Params: none. Example: call with no args → operations[{name,arity,description,angle_sensitive}].

When: need CODATA/math constant names (pi, e, qe, …) usable in evaluate. Params: none. Example: call with no args → constants[{name,value,unit,…}].

When: before convert_unit, or unknown conversion_id / unit pair. Params: none. Example: call with no args → conversions[{id,from,to,…}].

When: matrix/vector algebra (not infix evaluate). Params: op=add|sub|mul|transpose|det|inv|identity|ref|rref|eigen|dot|cross|norm|angle|unit; matrices=[A] or [A,B]; vector=[…] for norm/unit; n for identity; angle_mode for angle. Example: op="det", matrices=[[[1,2],[3,4]]].

When: one-variable summary stats, or STAT Norm Dist t/P/Q/R from a data list. Params: data (required); optional freq (same length); optional norm_x → adds t,P,Q,R (t=(x−mean)/σ_pop; P:−∞→t, Q:0→t, R:t→+∞). Example: data=[1,2,3,4,5], norm_x=4.

When: paired (x,y) stats and regression — not single-list stats_1var. Params: x,y equal length; model=linear|quadratic|logarithmic|exp|abexp|power|inverse| cubic|quartic|logistic|medmed (default linear); optional freq, predict_y_at, predict_x_at. Example: x=[1,2,3], y=[2,4,6], model="linear".

When: square linear system Ax=b (not polynomial roots or f(x)=0). Params: either A (n×n) and b (len n), or coefficients as augmented n×(n+1). Max n=32. Example: A=[[2,1],[1,3]], b=[1,2].

When: numeric root of infix f(x)=0 (prefer over guessing). Params: expression in x; prefer bracket=[a,b]; else guess; angle_mode for trig in f. Example: expression="x^2-2", bracket=[0,2].

When: roots of a polynomial a0+…+an x^n (degree 1–4), not general f(x). Params: coefficients=[a0,...,an]; allow_complex=bool (default true). Example: coefficients=[-2,0,1] for x^2-2=0.

When: convert an integer string between bases 2/8/10/16 (32-bit two's complement fixed). Params: value (digit string, no leading '-'; use FFFFFFFF-style for negatives); from_base, to_base in {2,8,10,16}. Example: value="FF", from_base=16, to_base=10.

When: integer/bitwise arithmetic in a chosen base (32-bit), not floating evaluate. Params: op=add|sub|mul|div|and|or|xor|xnor|not|neg; a; b (except not/neg); base=2|8|10|16. Example: op="add", a="A", b="5", base=16.

When: numerical derivative df/dx of infix f(x) at a point (not symbolic). Params: expression in x; at; angle_mode; optional h step. Example: expression="x^3", at=2.

When: numerical definite integral of infix f(x) on [lower, upper]. Params: expression in x; lower; upper; angle_mode; tol (default 1e-10). Example: expression="x^2", lower=0, upper=1.

When: discrete sum Σ f(x) for integer x from start to end inclusive. Params: expression in x; start; end; angle_mode. Example: expression="x+1", start=1, end=5.

When: discrete product Π f(x) for integer x from start to end inclusive. Params: expression in x; start; end; angle_mode. Example: expression="x", start=1, end=4 → 24.

When: prime factorization of a positive integer (not evaluate fact()). Params: n (positive integer, ≤10 digits). Example: n=12 → factors with multiplicity.

When: approximate minimum of infix f(x) on a closed interval. Params: expression in x; lower; upper; angle_mode; tol. Example: expression="(x-1)^2", lower=0, upper=2.

When: approximate maximum of infix f(x) on a closed interval. Params: expression in x; lower; upper; angle_mode; tol. Example: expression="-(x-1)^2", lower=0, upper=2.

When: convert rectangular (x,y) to polar (r,θ). Prefer over manual atan2 for this mode. Params: x, y; angle_mode for θ (default deg). Example: x=2, y=2, angle_mode="deg" → r≈2.828, θ=45.

When: convert polar (r,θ) to rectangular (x,y). Params: r, theta; angle_mode for θ (default deg). Example: r=2, theta=90, angle_mode="deg".

When: sexagesimal ° ′ ″ → decimal degrees. Params: degrees; optional minutes, seconds (default 0). Example: degrees=10, minutes=30, seconds=0 → 10.5.

When: decimal degrees → ° ′ ″ components. Params: decimal. Example: decimal=10.5 → degrees=10, minutes=30, seconds=0.

When: convert between listed measurement units (not free-form dimensional analysis). Params: value; either conversion_id OR from_unit+to_unit. Call list_unit_conversions first if unsure. Example: value=1, conversion_id="mile_to_km".

When: probability densities/CDFs/inverses (DISTR), or norm_p/q/r for standardized t. For Norm Dist from a data list use stats_1var(norm_x=…) instead. Params: type selects screen — pass every variable that type needs. Types: normal_pd (x,sigma,mu), normal_cd (lower,upper,sigma,mu), inverse_normal (area,sigma,mu,tail), binomial_pd/cd (x,n,p), inverse_binomial, poisson_*, geometric_*, t_pd/cd, chi2_pd/cd, f_pd/cd, norm_p/q/r (x=t). Example: type="normal_pd", x=36, sigma=2, mu=35.

When: show a real number in engineering form (significand + SI symbol). Prefer evaluate eng_symbols for expression results. Params: value. Example: value=12345 → display "12.345k".

When: ENG / ENG← style shift: multiply by 1000^steps (also engshift(x,n) in evaluate). Params: value; steps (default 1; negative shifts down). Example: value=1234, steps=1 → 1.234e6-style shift.

When: STAT hypothesis tests (not descriptive stats_1var / regression stats_2var). Params: type=z_test|t_test|2_samp_t_test|1_prop_z_test|2_prop_z_test|anova|linreg_ttest; pass editor fields for that type (data/sigma/mu0, x/n/p0, lists for ANOVA, …); alternative; pooled for two-sample. Example: type="t_test", data=[1,2,3], mu0=0.

When: LIST utilities (sequence, cumsum, sort, ΔList) — not stats summaries. Params: op=seq|cumsum|sort_a|sort_d|delta; seq needs expression,start,end[,step]; others need data; angle_mode for seq expressions. Example: op="cumsum", data=[1,2,3]; or op="seq", expression="2*x", start=1, end=3.

When: time-value-of-money (loan/annuity) — solve one of N,I,PV,PMT,FV. Params: solve_for=N|I|PV|PMT|FV; provide the other four; I is annual %; P_Y payments/year (default 1); C_Y compounds/year (default=P_Y); begin=True for BGN. Signs: outflow negative / inflow positive, kept consistent. Example: solve_for="PMT", N=12, I=6, PV=-1000, FV=0.

When: generate f(x) [and optional g(x)] values from start to end by step. Params: expression in x; start; end; step; optional expression2; angle_mode. Example: expression="2*x", start=0, end=2, step=1, expression2="x^2".

When: solve polynomial inequality a0+…+an x^n (degree 1–4). Params: coefficients=[a0,...,an]; relation=">"|">="|"<"|"<=". Example: coefficients=[-1,1], relation=">" for x-1>0.

When: proportion a:b = c:d with one unknown. Params: three of a,b,c,d known; solve_for=a|b|c|d|x (x = the single missing slot). Example: a=2, b=3, d=6, solve_for="c".